Past exam of the mathematics course of the University of Cambridge / 2021 / ib / Paper 4 / 6B (Numerical Analysis)

Use the nodes in the order $0,1,2,3$. Their first divided differences are $4,-2,6$, their second divided differences are $-3,4$, and their third divided difference is $7/3$. The Newton interpolation polynomial is thereforeThis is the unique interpolating polynomial of degree at most three through the four data points.

Appending the node $-2$ to the existing Newton interpolation polynomial givesAt the new node,The condition $p_4(-2)=10$ therefore givesHence